Calculus 2 — Premium Practice Exam

01 Integration by Parts

$$\int u\,dv = uv - \int v\,du$$

📌 LIATE Rule: Choose $u$ in order: Logarithm → Inverse trig → Algebraic → Trig → Exponential

✦ Quick Example $\int x e^x dx$: Let $u = x$, $dv = e^x dx$ ⟹ $du = dx$, $v = e^x$
$$= xe^x - \int e^x dx = xe^x - e^x + C$$

02 Trigonometric Substitution

$\sqrt{a^2 - x^2}$: let $x = a\sin\theta$
$\sqrt{a^2 + x^2}$: let $x = a\tan\theta$
$\sqrt{x^2 - a^2}$: let $x = a\sec\theta$

📌 Key: After substitution, use Pythagorean identities to simplify. Always convert back to $x$.

✦ Quick Example $\int \frac{dx}{\sqrt{4-x^2}}$: Let $x = 2\sin\theta$, $dx = 2\cos\theta\,d\theta$
$= \int \frac{2\cos\theta}{2\cos\theta}\,d\theta = \theta + C = \arcsin\!\left(\tfrac{x}{2}\right) + C$

03 Partial Fraction Decomposition

$\dfrac{P(x)}{(x-a)(x-b)} = \dfrac{A}{x-a} + \dfrac{B}{x-b}$

Irreducible quadratic: $\dfrac{Ax+B}{x^2+bx+c}$

📌 Key: Degree of numerator must be less than denominator. If not, do polynomial long division first.

✦ Quick Example $\int \frac{1}{x^2-1}dx = \int \frac{1}{(x-1)(x+1)}dx = \frac{1}{2}\int\!\left(\frac{1}{x-1}-\frac{1}{x+1}\right)dx = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right|+C$

04 Improper Integrals

$\int_a^{\infty} f(x)\,dx = \lim_{t\to\infty}\int_a^t f(x)\,dx$

📌 p-integral: $\int_1^{\infty} \frac{1}{x^p}dx$ converges if $p > 1$, diverges if $p \le 1$

✦ Quick Example $\int_1^{\infty} \frac{1}{x^2}dx = \lim_{t\to\infty}\left[-\frac{1}{x}\right]_1^t = \lim_{t\to\infty}\left(1-\frac{1}{t}\right) = 1$

05 Arc Length & Surface Area

Arc Length: $L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$

Surface Area (rotation about x-axis): $S = 2\pi\int_a^b f(x)\sqrt{1+[f'(x)]^2}\,dx$

📌 Key: For parametric: $L = \int_\alpha^\beta \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$

06 Sequences & Series Convergence

Geometric Series: $\sum_{n=0}^{\infty} ar^n = \dfrac{a}{1-r}$, $|r| < 1$

p-Series: $\sum_{n=1}^{\infty} \dfrac{1}{n^p}$ converges iff $p > 1$

📌 Divergence Test: If $\lim_{n\to\infty} a_n \neq 0$, the series diverges.
📌 Telescoping: Write partial sums and cancel terms.

07 Ratio & Root Tests

Ratio Test: $L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$

Root Test: $L = \lim_{n\to\infty}\sqrt[n]{|a_n|}$

$L < 1$: converges absolutely · $L > 1$: diverges · $L = 1$: inconclusive

📌 Best for: Ratio test → factorials & exponentials. Root test → $n$th powers.

08 Power Series & Taylor/Maclaurin Series

Taylor: $f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$

Maclaurin ($a=0$):
$e^x = \sum\frac{x^n}{n!}$, $\;\sin x = \sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}$, $\;\cos x = \sum\frac{(-1)^n x^{2n}}{(2n)!}$
$\frac{1}{1-x} = \sum x^n$, $\;\ln(1+x) = \sum\frac{(-1)^{n+1}x^n}{n}$

📌 Radius of Convergence: Use Ratio Test to find $R$. Interval: $(a-R, a+R)$; check endpoints.

09 Polar Coordinates

Area: $A = \dfrac{1}{2}\int_\alpha^\beta r^2\,d\theta$

Conversions: $x = r\cos\theta$, $y = r\sin\theta$, $r^2 = x^2 + y^2$

📌 Key Curves: $r = a$ (circle), $r = 2a\cos\theta$ (circle), $r = 1+\cos\theta$ (cardioid), $r = a\sin(n\theta)$ (rose with $n$ petals if $n$ odd, $2n$ if $n$ even)

10 Alternating Series & Absolute Convergence

Alternating Series Test (Leibniz): $\sum (-1)^n b_n$ converges if
(1) $b_n > 0$ is decreasing, and (2) $\lim_{n\to\infty} b_n = 0$

📌 Error bound: $|S - S_N| \le b_{N+1}$
📌 Absolute convergence implies convergence. Converse is false.

Calculus II

Concept Review & Key Formulas